Abstract We present a new probabilistic analysis of distributed systems. Our approach relies on the theory of quasi-stationary distributions (QSD) and the results recently developed by the first and third authors. We give properties on the deadlock time and the distribution of the model before deadlock, both for discrete and diffusion models. Our results apply to any finite values of the involved parameters

Abstract A theoretical approach for solving time-fractional stochastic Ginzburg–Landau equation with mixed fractional Brownian motion in Hilbert space is elaborated. Initially, the stochastic partial differential system is reformulated in the Hilbert space by using the properties of fractional order space and fractional Laplacian. We establish the existence of mild solutions by employing Mittag–Leffler

Abstract In this paper, a Haar wavelet method is proposed for solving linear and nonlinear stochastic Itô–Volterra integral equation (SIVIE) driven by fractional Brownian motion (FBM) with Hurst parameter H ∈ ( 1 2 , 1 ) . This approach reduces the solution of the problem under study to the solution of a linear and nonlinear system of algebraic equations, which is based on the operational matrix and

Abstract In this article, we investigate a zero-sum stochastic game for continuous-time Markov chain with denumerable state space and unbounded transition rates, under the probability criterion. Under suitable assumptions, we show the existence of value of the game and also characterize it as the unique solution of a pair of Shapley equations. We also establish the existence of a randomized stationary

Abstract We generalize the Kolmogorov continuity theorem and prove the continuity of a class of stochastic fields with the parameter. As an application, we derive the continuity of solutions for nonlocal stochastic parabolic equations driven by non-Gaussian Lévy noises.

Abstract We consider an investor whose portfolio consists of a single risky asset and a risk free asset. The risky asset’s return has a heavy tailed distribution and thus does not have higher order moments. Hence, she aims to maximize the expected utility of the portfolio defined in terms of the median return. This is done subject to managing the Value at Risk (VaR) defined in terms of a high order

Abstract The mean time to finish a project is studied where the rate of work varies randomly. It is proved that the mean time to finish is bounded below by the time taken to complete the project if the work is performed at a fixed average rate, specifically, at the average work rate of the corresponding unrestricted or non-exit problem. Several examples illustrate the usefulness of the result in a

Abstract The main goal of this work is to relate weak and pathwise mild solutions for parabolic quasilinear stochastic partial differential equations (SPDEs). Extending in a suitable way techniques from the theory of nonautonomous semilinear SPDEs to the quasilinear case, we prove the equivalence of these two solution concepts.

Abstract Contributions of the present paper consist of two parts. In the first one, we contribute to the theory of stochastic calculus for signed measures. For instance, we provide some results permitting to characterize martingales and Brownian motion both defined under a signed measure. We also prove that the uniformly integrable martingales (defined with respect to a signed measure) can be expressed

Abstract We provide a probabilistic SIRD model for the COVID-19 pandemic in Italy, where we allow the infection, recovery and death rates to be random. In particular, the underlying random factor is driven by a fractional Brownian motion. Our model is simple and needs only some few parameters to be calibrated.

Abstract The success in reducing global HIV prevalence rates is attributed to control measures like information and education campaigns (IECs), antiretroviral therapy (ART), and national, multinational and multilateral support providing official developmental assistance (ODAs) to combat HIV. A class of stochastic nonlinear multi-population behavioral HIV/AIDS models is studied, where behavioral change

Abstract We give a particle picture interpretation of two recently discovered classes of self-similar stable processes with stationary increments studied by Samorodnitsky et al. and Dombry and Guillotin-Plantard. We study the occupation times of certain Poissonian systems of particles with ±1 charges and i.i.d. heavy-tailed weights, moving independently according to Lévy processes. We also obtain a

Abstract In this article, we study risk-sensitive stochastic differential games for controlled reflecting diffusion processes in a smooth bounded domain. We analyze the ergodic cost evaluation criterion for both nonzero-sum games and zero-sum games. Using principal eigenvalue approach, we establish the existence of Nash/saddle-point equilibria for relevant cases.

Abstract We study the strong convergence of the Carathéodory numerical scheme for a class of nonlinear McKean-Vlasov stochastic differential equations (MVSDE). We prove, under Lipschitz assumptions, the convergence of the approximate solutions to the unique solution of the MVSDE. Moreover, we show that the result remains valid, under continuous coefficients, provided that pathwise uniqueness holds

Abstract The first-passage time τ S ( x ) of a one-dimensional continuous stochastic process X ( t ) , starting from x ≤ S ( 0 ) , through a smooth boundary S(t) is investigated; in particular, diffusions and some kinds of Gaussian processes, such as Gauss-Markov and their fractional integrals, are considered. The tail behavior of P ( max s ∈ [ 0 , t ] X ( s ) > R ) and related asymptotics for τ S

Abstract Within the rough path framework, we prove the continuity of the solution to random differential equations driven by a one dimensional fractional Brownian motion with respect to the Hurst parameter H when H ∈ ( 1 / 3 , 1 / 2 ] .

Abstract The global well-posedness as well as long-term behavior in terms of mean random attractors and invariant measures are investigated for a class of stochastic discrete reaction-diffusion equations defined on Z k with a family of superlinear noise. The existence and uniqueness of weak pullback mean random attractors for the mean random dynamical system associated with the non-autonomous equations

Abstract The article is devoted to the existence and Hyers-Ulam stability of the almost periodic solution to the fractional differential equation with impulse and fractional Brownian motion under nonlocal condition. The investigation is mainly based on the semigroups of operators method and Mönch fixed points method, as well as the basic theory of Hyers-Ulam stability.

Abstract We study the asymptotic behavior of a real-valued diffusion whose nonregular drift is given as a sum of a dissipative term and a bounded measurable one. We prove that two trajectories of that diffusion converge almost sure to one another at an exponential explicit rate as soon as the dissipative coefficient is large enough. A similar result in Lp is obtained.

In this paper, we discuss a class of stochastic nonlinear partial differential equation of Burgers type driven by pseudo differential operator ( Δ + Δ α ) for α ∈ ( 0 , 2 ) , and fractional Brownian sheet. The existence and uniqueness of an Lp-valued (local) solution is established for the initial valued problem to the equation.

We investigate the joint distribution and the multivariate survival functions for the maxima of an Ornstein-Uhlenbeck (OU) process in consecutive time-intervals. A PDE method, alongside an eigenfunction expansion is adopted, with which we first calculate the distribution and the survival functions for the maximum of a homogeneous OU-process in a single interval. By a deterministic time-change and a

The aim of this manuscript is to analyze the existence of mild solution of non-instantaneous impulsive Hilfer fractional stochastic differential equations (NIHFSDEs) driven by fractional Brownian motion (fBm). Sufficient conditions for a class of NIHFSDEs of order 0 < β < 1 and of type 0 ≤ α ≤ 1 driven by fBm is derived with the help of fractional calculus, stochastic theory, fixed point theorem and

In the paper, we consider some piecewise deterministic Markov process, whose continuous component evolves according to semiflows, which are switched at the jump times of a Poisson process. The associated Markov chain describes the states of this process directly after the jumps. Certain ergodic properties of these two dynamical systems have been already investigated in our recent papers. We now aim

We introduce and study a marked Poisson random measure on a σ-compact Hausdorff space. The underlying parameters of this measure are changing in accordance with the evolution of some stochastic process. This random measure (also known as a modulated measure) resembles those of the conventional Poisson random measure. Among many applications, the one about Poisson random measures modulated by semi-Markov

Let W denote the Brownian motion. For any exponentially bounded Borel function g the function u defined by u ( t , x ) = E [ g ( x + σ W T − t ) ] is the stochastic solution of the backward heat equation with terminal condition g. Let u n ( t , x ) denote the corresponding approximation generated by a simple symmetric random walk with time steps 2 T / n and space steps ± σ T / n where σ > 0 . For a

Kolmogorov’s converse inequality is proved for dependent random variables in order to obtain necessary and sufficient conditions for weak and strong laws (for not necessarily strictly stationary sequences) under strong dependence without rate assumptions.

Large deviation principle by the weak convergence approach is established for the stochastic nonlinear Schrödinger equation in one-dimension and as an application the exit problem is investigated.

We consider a new class of one-dimensional diffusion processes with self-similar random potentials. The self-similar random potential has different exponents to the left and the right hand sides of the origin. We show that, because of the difference between the two exponents, the long-time behaviors of our process on the left and the right hand sides of the origin are quite different from each other

On the global platform of emerging infectious diseases, dengue fever added a serious health concern, especially in the tropical and subtropical countries with poor health services. Due to unavailability of proper vaccination, primary prevention of dengue is possible at social and personal levels by controlling the propagation of vectors as well as avoiding the bites of infected ones, respectively.

Abstract We formulate ill-posedness of inverse problems of estimation and prediction of Coronavirus Disease 2019 (COVID-19) outbreaks from statistical and mathematical perspectives. This is by nature a stochastic problem, since e.g., random perturbation in parameters can cause instability of estimation and prediction. This leaves us with a plenty of possible statistical regularizations, thus generating

This paper is concerned with existence, uniqueness, and approximate controllability of mild solutions to second-order non-autonomous stochastic impulsive differential systems. The impulsive differential systems may provide mathematical models to the phenomena having discontinuous jumps. The main results are carried out by using the generalized Banach contraction principle, evolution family, and stochastic

We study optimal control for mean-field forward–backward stochastic differential equations with payoff functionals of mean-field type. Sufficient and necessary optimality conditions in terms of a stochastic maximum principle are derived. As an illustration, we solve an optimal portfolio with mean-field risk minimization problem.

We prove an anticipative sufficient stochastic minimum principle in a jump process setup with initially enlarged filtrations. We apply the result to several portfolio selection problems like mean and minimal variance hedging under enlarged filtrations. We also investigate utility maximizing portfolio selection under future information. Contrarily to classical optimization methods like dynamic programing

In this article, we derive the risk-neutral measures of the stock options price and volatility by incorporating a simple constrained minimization of the Kullback measure of relative information. We obtain a second-order parabolic partial differential equation, the generalized Black–Scholes equation based on the theoretical analysis when the underlying financial asset is estimated using a stochastic

In this paper, stochastic integral equations characterized by fractional Brownian motion have been studied. The fractional stochastic integral equation has been solved by second kind Chebyshev wavelets. The convergence and error analysis have been discussed for the efficiency of the discussed method. In addition, two illustrative examples have been solved to examine the efficiency and accuracy of the

In this paper, by using Girsanov’s transformation and the property of the corresponding reference stochastic differential equations, we investigate weak existence and uniqueness of solutions and weak convergence of Euler-Maruyama scheme to stochastic functional differential equations with Hölder continuous drift driven by fractional Brownian motion with Hurst index H ∈ ( 1 / 2 , 1 ) .

We study nonzero-sum stochastic differential games with risk-sensitive discounted cost criteria. Under fairly general conditions on drift term and diffusion coefficients, we establish a Nash equilibrium in Markov strategies for the discounted cost criterion. We achieve our results by studying relevant systems of coupled HJB equations.

This paper considers an optimal time-consistent proportional reinsurance problem with constraints on the strategies under the mean-variance criterion for an insurer. It is assumed that the surplus process is described by a compound Poisson risk model with dependent classes of insurance business named thinning-dependence structure, in which the stochastic sources related to claim occurrence are classified

Abstract In this correspondence, for an array of rowwise independent random elements {Vn,k,1≤k≤kn,n≥1,kn→∞} taking values in a real separable Rademacher type p (1≤p≤2) Banach space and a sequence of positive constants {cn,n≥1}, the main result provides conditions for the complete convergence result ∑n=1∞cnP(max1≤k≤kn||∑i=1kVn,i||>ε)<∞ for all ε>0 to hold. The complete convergence does not necessary

Abstract The objective of this paper is to investigate the existence of mild solutions and optimal controls for a class of fractional neutral stochastic differential equations (NSDEs) driven by fractional Brownian motion and Poisson jumps in Hilbert spaces. First, we establish a new set of sufficient conditions for the existence of mild solutions of the aforementioned fractional systems by using the

Abstract This paper explores the relationship between non-Markovian fully coupled forward–backward stochastic systems and path-dependent PDEs. The definition of classical solution for the path-dependent PDE is given within the framework of functional Itô calculus. Under mild hypotheses, we prove that the forward–backward stochastic system provides the unique classical solution to the path-dependent

Abstract The present paper addresses the issue of choosing an optimal dynamic reinsurance policy, which is state-dependent, for an insurance company that operates under multiple insurance business lines. For each line, the Cramer–Landberg model is adopted for the risk process and one of the contracts such as Proportional reinsurance, excess-of-loss reinsurance (XL) and limited XL reinsurance (LXL)

Abstract Multivariate Bessel processes (Xt,k)t≥0 describe interacting particle systems of Calogero-Moser-Sutherland type and are related with β-Hermite and β-Laguerre ensembles. They depend on a root system and a multiplicity k. Recently, several limit theorems were derived for k→∞ with fixed starting point. Moreover, the SDEs of (Xt,k)t≥0 were used to derive strong laws of large numbers for k→∞ with

Abstract After establishing the moderate deviation principle by the classical Azencott method, we prove the Strassen’s compact law of the iterated logarithm (LIL) for a class of stochastic partial differential equations (SPDEs). As an application, we obtain this type of LIL for two population models known as super-Brownian motion and Fleming-Viot process. In addition, the classical LIL is shown for

Abstract We propose a local linearization scheme to approximate the solutions of non-autonomous stochastic differential equations driven by fractional Brownian motion with Hurst parameter 1/2

Abstract In this article, based on the tool of coupling method, Girsanov’s theorem and a stopping argument, we establish the exponential ergodicity for white noise driven parabolic stochastic partial differential equations when the coefficients are non-Lipschitz continuous.

Abstract The evolution of biofilms on surfaces of medical implants, food, machinery, etc. may be modeled via reaction-diffusion partial differential equations. According to Allee effect, the microbes growth rate is positively correlated with the microbes density. Thus, instead of using Fisher’s equation, we employ the bistable Allen–Cahn equation for the modeling of biofilm formation. We work in space

Abstract Many engineering materials of interest are polycrystals: an aggregate of many crystals with size usually below 100 μm. Those small crystals are called the grains of the polycrystal, and are often equiaxed. However, because of processing, the grain shape may become anisotropic; for instance, during recrystallization or phase transformations, the new grains may grow in the form of ellipsoids

Abstract The goal of this paper is to establish the Lipschitz and W2,∞ estimates for a second-order parabolic PDE ∂tu(t,x)=12Δu(t,x)+f(t,x) on Rd with zero initial data and f satisfying a Ladyzhenskaya–Prodi–Serrin type condition. Following the theoretic result, we then give two applications. The first is to discuss the regularity of the stochastic heat equations, and the second is to discuss the Sobolev

Abstract We study an inverse problem for the first-passage place of a one-dimensional diffusion process X(t) (also with jumps), starting from a random position η∈[a,b]. Let be τa,b the first time at which X(t) exits the interval (a,b), and πa=P(X(τa,b)≤a) the probability of exit from the left of (a,b). Given a probability q∈(0,1), the problem consists in finding the density g of η (if it exists) such

Abstract We propose a nonparametric estimation of random effects from the following small fractional diffusions dXi(t)=βib(Xi(t))dt+εdWi,H(t), Xi(0)=x0i, 0≤t≤T. For i=1,⋯,N, βi is random variable and (Wi,H(t), t≥0, i=1,⋯,N) is are standard fractional Brownian motions with a common known Hurst index H∈(1/2,1). The asymptotic behavior of the proposed estimators is established when ε and N are, respectively

Abstract We study the locally uniform convergence from a family of pullback random attractors to a deterministic attractor. We establish criteria by using joint convergence of the cocycles, collective local compactness and deterministic recurrence of the random attractors. All three conditions are verified in the content of stochastic non-autonomous Magneto-hydrodynamics equations under the weak assumption

Abstract The Adomian decomposition method (ADM) is a powerful tool to solve several nonlinear functional equations and a large class of initial/boundary value problems. In this paper, we discuss a probabilistic approach to compute the Adomian polynomials (AP’s), which is the main part of the ADM. We provide a probabilistic interpretation for the AP’s, both for the one-variable and the multivariable

Abstract This paper deals with the limiting behavior of non-autonomous stochastic reaction-diffusion equations without uniqueness on unbounded narrow domains. We prove the existence and upper semicontinuity of random attractors for the equations on a family of unbounded (n + 1)-dimensional narrow domains, which collapses onto an n-dimensional domain. Since the solutions are non-uniqueness, which leads

Abstract We provide conditions implying finite-time blowup of positive weak solutions to the semilinear equation du(t,x)=[Δαu(t,x)+Ku(t,x)+u1+β(t,x)]dt+μu(t,x) dBtH,u(0,x)=f(x),x∈R d,t≥0, where α∈(0,2],K∈R,β>0,μ≥0 and H∈[12,1) are constants, Δα is the fractional power −(−Δ)α/2 of the Laplacian, (BtH) is a fractional Brownian motion with Hurst parameter H, and f≥0 is a bounded measurable function. To

Abstract In this paper, we consider a stochastic tuberculosis model with two kinds of treatments, that is, treatment at home and treatment in hospital. Firstly, we obtain sufficient conditions for extinction and persistence in the mean of the diseases, then in the case of persistence, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive

Abstract Complementary regularity between the integrand and integrator is a well known condition for the integral ∫0Tf(r) dg(r) to exist in the Riemann-Stieltjes sense. This condition also applies to the multi-dimensional case, in particular the 2 D integral ∫[0,T]2f(s,t) dg(s,t). In the paper, we give a new condition for the existence of the integral under the assumption that the integrator g is a

In this article, we derive the state probabilities of different type of space- and time-fractional Poisson processes using z-transform. We work on tempered versions of time-fractional Poisson process and space-fractional Poisson processes. We also introduce Gegenbauer type fractional differential equations and their solutions using z-transform. Our results generalize and complement the results available

In the present paper we consider a family of non-Volterra quadratic stochastic operators depending on a parameter α and study their trajectory behaviors. We find all fixed and periodic points for a non-Volterra quadratic stochastic operator on a finite-dimensional simplex. A complete description of the set of limit points is given, and we show that such operators have the ergodic property.

The aim of this article is to consider the existence, uniqueness and uniformly exponential p-stability of the solution for ∇-delay stochastic dynamic equations on time scales via Lyapunov functions. This work can be considered as a unification and generalization of stochastic difference and stochastic differential time-varying delay equations.